exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 9

Speedup: 0.4×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-(1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-(1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-1}}{{\left(e^{x}\right)}^{\left(-x\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp -1.0) (pow (exp x) (- x))))

C

double code(double x) {
	return exp(-1.0) / pow(exp(x), -x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((-1.0d0)) / (exp(x) ** -x)
end function

Java

public static double code(double x) {
	return Math.exp(-1.0) / Math.pow(Math.exp(x), -x);
}

Python

def code(x):
	return math.exp(-1.0) / math.pow(math.exp(x), -x)

Julia

function code(x)
	return Float64(exp(-1.0) / (exp(x) ^ Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = exp(-1.0) / (exp(x) ^ -x);
end

Wolfram

code[x_] := N[(N[Exp[-1.0], $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   3. neg-sub0 N/A

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   4. lift--.f64 N/A

      \[\leadsto e^{0 - \color{blue}{\left(1 - x \cdot x\right)}} \]

   5. sub-neg N/A

      \[\leadsto e^{0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}} \]

   6. associate--r+ N/A

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) - \left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   7. metadata-eval N/A

      \[\leadsto e^{\color{blue}{-1} - \left(\mathsf{neg}\left(x \cdot x\right)\right)} \]

   8. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{-1}}{e^{\mathsf{neg}\left(x \cdot x\right)}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{-1}}{e^{\mathsf{neg}\left(x \cdot x\right)}}} \]

   10. lower-exp.f64 N/A

       \[\leadsto \frac{\color{blue}{e^{-1}}}{e^{\mathsf{neg}\left(x \cdot x\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{e^{-1}}{e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)}} \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \frac{e^{-1}}{e^{\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}}} \]

   13. exp-prod N/A

       \[\leadsto \frac{e^{-1}}{\color{blue}{{\left(e^{x}\right)}^{\left(\mathsf{neg}\left(x\right)\right)}}} \]

   14. lower-pow.f64 N/A

       \[\leadsto \frac{e^{-1}}{\color{blue}{{\left(e^{x}\right)}^{\left(\mathsf{neg}\left(x\right)\right)}}} \]

   15. lower-exp.f64 N/A

       \[\leadsto \frac{e^{-1}}{{\color{blue}{\left(e^{x}\right)}}^{\left(\mathsf{neg}\left(x\right)\right)}} \]

   16. lower-neg.f64 100.0

       \[\leadsto \frac{e^{-1}}{{\left(e^{x}\right)}^{\color{blue}{\left(-x\right)}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{-1}}{{\left(e^{x}\right)}^{\left(-x\right)}}} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x \cdot x + -1} \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp (+ (* x x) -1.0)) 0.5) (/ 1.0 (E)) (/ (* x x) (E))))

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x)))) < 0.5

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

      9. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

      11. lower-exp.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

      12. exp-1-e N/A

          \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

      13. lower-E.f64 100.0

          \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{E}\left(\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{E}\left(\right)} \]

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x))))

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

      9. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

      11. lower-exp.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

      12. exp-1-e N/A

          \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

      13. lower-E.f64 100.0

          \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

      3. lower-fma.f64 49.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   7. Applied rewrites 49.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{\color{blue}{2}}}{\mathsf{E}\left(\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.3%

       \[\leadsto \frac{x \cdot \color{blue}{x}}{\mathsf{E}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x \cdot x + -1} \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-6}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right) \cdot \mathsf{E}\left(\right), x \cdot x, \mathsf{E}\left(\right)\right), x \cdot x, \mathsf{E}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e-6)
   (/
    1.0
    (fma
     (- (fma (* (fma (* 0.16666666666666666 x) x -0.5) (E)) (* x x) (E)))
     (* x x)
     (E)))
   (exp (* x x))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999955e-7

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

      9. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

      11. lower-exp.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

      12. exp-1-e N/A

          \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

      13. lower-E.f64 100.0

          \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

      3. lower-fma.f64 99.8

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(x, x, 1\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(x, x, 1\right)}}} \]

      4. lower-/.f64 99.8

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(x, x, 1\right)}}} \]

   9. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(x, x, 1\right)}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right) + \left(\frac{-1}{2} \cdot \mathsf{E}\left(\right) + \frac{1}{6} \cdot \mathsf{E}\left(\right)\right)\right)\right) - \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right)\right) - \mathsf{E}\left(\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right) + \left(\frac{-1}{2} \cdot \mathsf{E}\left(\right) + \frac{1}{6} \cdot \mathsf{E}\left(\right)\right)\right)\right) - \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right)\right) - \mathsf{E}\left(\right)\right) + \mathsf{E}\left(\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\left({x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right) + \left(\frac{-1}{2} \cdot \mathsf{E}\left(\right) + \frac{1}{6} \cdot \mathsf{E}\left(\right)\right)\right)\right) - \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right)\right) - \mathsf{E}\left(\right)\right) \cdot {x}^{2}} + \mathsf{E}\left(\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right) + \left(\frac{-1}{2} \cdot \mathsf{E}\left(\right) + \frac{1}{6} \cdot \mathsf{E}\left(\right)\right)\right)\right) - \left(-1 \cdot \mathsf{E}\left(\right) + \frac{1}{2} \cdot \mathsf{E}\left(\right)\right)\right) - \mathsf{E}\left(\right), {x}^{2}, \mathsf{E}\left(\right)\right)}} \]

   12. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{E}\left(\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right), x \cdot x, \mathsf{E}\left(\right)\right), x \cdot x, \mathsf{E}\left(\right)\right)}} \]

   if 9.99999999999999955e-7 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto e^{\color{blue}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

      2. lower-*.f64 99.4

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 99.4%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-6}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right) \cdot \mathsf{E}\left(\right), x \cdot x, \mathsf{E}\left(\right)\right), x \cdot x, \mathsf{E}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(x - 1, x, x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (fma (- x 1.0) x (- x 1.0))))

C

double code(double x) {
	return exp(fma((x - 1.0), x, (x - 1.0)));
}

Julia

function code(x)
	return exp(fma(Float64(x - 1.0), x, Float64(x - 1.0)))
end

Wolfram

code[x_] := N[Exp[N[(N[(x - 1.0), $MachinePrecision] * x + N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   2. neg-sub0 N/A

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   3. lift--.f64 N/A

      \[\leadsto e^{0 - \color{blue}{\left(1 - x \cdot x\right)}} \]

   4. associate--r- N/A

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

   5. metadata-eval N/A

      \[\leadsto e^{\color{blue}{-1} + x \cdot x} \]

   6. +-commutative N/A

      \[\leadsto e^{\color{blue}{x \cdot x + -1}} \]

   7. lift-*.f64 N/A

      \[\leadsto e^{\color{blue}{x \cdot x} + -1} \]

   8. lower-fma.f64 100.0

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto e^{\color{blue}{x \cdot x + -1}} \]

   2. difference-of-sqr--1 N/A

      \[\leadsto e^{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]

   3. *-commutative N/A

      \[\leadsto e^{\color{blue}{\left(x - 1\right) \cdot \left(x + 1\right)}} \]

   4. distribute-lft-in N/A

      \[\leadsto e^{\color{blue}{\left(x - 1\right) \cdot x + \left(x - 1\right) \cdot 1}} \]

   5. lower-fma.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x - 1, x, \left(x - 1\right) \cdot 1\right)}} \]

   6. lower--.f64 N/A

      \[\leadsto e^{\mathsf{fma}\left(\color{blue}{x - 1}, x, \left(x - 1\right) \cdot 1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto e^{\mathsf{fma}\left(x - 1, x, \color{blue}{\left(x - 1\right) \cdot 1}\right)} \]

   8. lower--.f64 100.0

      \[\leadsto e^{\mathsf{fma}\left(x - 1, x, \color{blue}{\left(x - 1\right)} \cdot 1\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x - 1, x, \left(x - 1\right) \cdot 1\right)}} \]

7. Final simplification 100.0%

   \[\leadsto e^{\mathsf{fma}\left(x - 1, x, x - 1\right)} \]
8. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (fma x x -1.0)))

C

double code(double x) {
	return exp(fma(x, x, -1.0));
}

Julia

function code(x)
	return exp(fma(x, x, -1.0))
end

Wolfram

code[x_] := N[Exp[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   2. neg-sub0 N/A

      \[\leadsto e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]

   3. lift--.f64 N/A

      \[\leadsto e^{0 - \color{blue}{\left(1 - x \cdot x\right)}} \]

   4. associate--r- N/A

      \[\leadsto e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]

   5. metadata-eval N/A

      \[\leadsto e^{\color{blue}{-1} + x \cdot x} \]

   6. +-commutative N/A

      \[\leadsto e^{\color{blue}{x \cdot x + -1}} \]

   7. lift-*.f64 N/A

      \[\leadsto e^{\color{blue}{x \cdot x} + -1} \]

   8. lower-fma.f64 100.0

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\frac{x}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right)\right) \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (* (/ x (E)) (* x x)) x)
  (fma (* 0.16666666666666666 x) x 0.5)
  (/ (fma x x 1.0) (E))))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   3. exp-neg N/A

      \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

   5. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

   9. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

   12. exp-1-e N/A

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   13. lower-E.f64 100.0

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)} + \frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) + \frac{1}{\mathsf{E}\left(\right)}\right) + \frac{1}{\mathsf{E}\left(\right)}} \]

7. Applied rewrites 89.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{x}^{4}}{\mathsf{E}\left(\right)}, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 89.7%

   \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\mathsf{E}\left(\right)}, \mathsf{fma}\left(\color{blue}{0.16666666666666666} \cdot x, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \]
10. Step-by-step derivation

11. Applied rewrites 89.7%

    \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{x}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(\color{blue}{0.16666666666666666 \cdot x}, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \]

12. Final simplification 89.7%

    \[\leadsto \mathsf{fma}\left(\left(\frac{x}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right)\right) \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \]
13. Add Preprocessing

Alternative 7: 91.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right), \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{1}{\mathsf{E}\left(\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (/ (* x x) (E)) (* x x))
  (fma (* 0.16666666666666666 x) x 0.5)
  (/ 1.0 (E))))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   3. exp-neg N/A

      \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

   5. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

   9. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

   12. exp-1-e N/A

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   13. lower-E.f64 100.0

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{x}^{2}}{\mathsf{E}\left(\right)} + \frac{1}{2} \cdot \frac{1}{\mathsf{E}\left(\right)}\right) + \frac{1}{\mathsf{E}\left(\right)}\right) + \frac{1}{\mathsf{E}\left(\right)}} \]

7. Applied rewrites 89.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{x}^{4}}{\mathsf{E}\left(\right)}, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 89.7%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{x \cdot x}{\mathsf{E}\left(\right)}, \mathsf{fma}\left(\color{blue}{0.16666666666666666 \cdot x}, x, 0.5\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{x \cdot x}{\mathsf{E}\left(\right)}, \mathsf{fma}\left(\frac{1}{6} \cdot x, x, \frac{1}{2}\right), \frac{1}{\mathsf{E}\left(\right)}\right) \]
11. Step-by-step derivation

12. Applied rewrites 89.2%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{x \cdot x}{\mathsf{E}\left(\right)}, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{1}{\mathsf{E}\left(\right)}\right) \]

13. Final simplification 89.2%

    \[\leadsto \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{E}\left(\right)} \cdot \left(x \cdot x\right), \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 0.5\right), \frac{1}{\mathsf{E}\left(\right)}\right) \]
14. Add Preprocessing

Alternative 8: 76.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x x 1.0) (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   3. exp-neg N/A

      \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

   5. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

   9. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

   12. exp-1-e N/A

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   13. lower-E.f64 100.0

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

   3. lower-fma.f64 72.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

7. Applied rewrites 72.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]
8. Add Preprocessing

Alternative 9: 52.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{-\left(1 - x \cdot x\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}} \]

   3. exp-neg N/A

      \[\leadsto \color{blue}{\frac{1}{e^{1 - x \cdot x}}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{1 - x \cdot x}}} \]

   5. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{x \cdot x}}}} \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{e^{1}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{x \cdot x}}}{e^{1}} \]

   9. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e^{1}} \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(e^{x}\right)}}^{x}}{e^{1}} \]

   12. exp-1-e N/A

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

   13. lower-E.f64 100.0

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{E}\left(\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{E}\left(\right)} \]
6. Step-by-step derivation

7. Applied rewrites 47.3%

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{E}\left(\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))